Sound Radiation from a Buried Nozzle with Jet and Bypass Flow

Radiation of sound from a simplified model of a buried nozzle with bypass flow is studied. More precisely, a semi-infinite duct (the inner nozzle) is situated inside a larger semi-infinite duct. The exit plane of the larger duct either coincides with the exit plane of the smaller duct, or extends beyond it. Differences in the piece-wise subsonic mean flow velocity, density and temperature are taken into account. The inner nozzle issues the core flow inside the bypass jet flow. The bypass nozzle issues the bypass jet flow inside the ambient co-flow. Two vortex sheets, attached to the duct exits, separate the different flows from each other. These vortex sheets are unstable due to this mean velocity discontinuity. The application of the Kutta condition at the respective trailing edges guarantees shedding of vorticity which excites these instabilities. The system is set up to respond to an incident annular duct mode, but the analysis would be very similar for an inner duct mode. To obtain an analytical solution a Wiener-Hopf approach with Idemen’s method of “weak factorisation” is applied. Formulation of the boundary value problem following the classical approach leads to a couple of simultaneous Wiener-Hopf equations. These equations produce a matrix equation system, which is decoupled by the introduction of an infinite sum of poles. The uncoupled scalar equations are solved independently by a standard application of analytical continuation. The final solution includes unknown coefficients which are determined by solving an infinite linear algebraic system numerically. The contribution of the instability waves are separated from the rest of the solution. The asymptotic far field is found by a standard application of the steepest descent method. Finally a series of practical examples are given.